Механика.Методика решения задач
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Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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II. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ ɢ ɜɵɛɪɚɧɧɵɦɢ ɦɨɞɟɥɹɦɢ ɬɟɥɚ ɢ ɟɝɨ ɞɜɢɠɟɧɢɹ ɡɚɩɢɲɟɦ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɫɤɨɪɟɧɢɹ ɬɟɥɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:
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(1.40) |
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III. Ɂɚɩɢɫɚɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɟɤɰɢɣ ɫɤɨɪɨɫɬɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɡɧɚɱɟ-
ɧɢɣ ɩɨɡɜɨɥɹɸɬ ɧɚɣɬɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ȣ(t) |
ɢ ɡɚɤɨɧ |
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ɟɝɨ ɞɜɢɠɟɧɢɹ r(t) ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ: |
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ɍɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɧɚɯɨɞɢɬɫɹ ɢɡ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ (1.43) ɩɭɬɟɦ ɢɫɤɥɸɱɟɧɢɹ ɜɪɟɦɟɧɢ t:
gx2
y(x) H . (1.44)
2X02
Ɉɫɬɚɥɶɧɵɟ ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɚɦɢ, ɩɪɢɜɟɞɟɧɧɵɦɢ ɜ ɩ. 1 ɞɚɧɧɨɣ Ƚɥɚɜɵ.
Ɇɨɞɭɥɶ ɫɤɨɪɨɫɬɢ (1.8) ɪɚɜɟɧ:
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Xx2 |
Xy2 |
X02 (gt)2 . |
(1.45) |
Ɇɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ (1.14) ɢɦɟɟɬ ɜɢɞ: |
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ɉɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɫɤɨɪɨɫɬɢ ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɟ ɟɦɭ ɧɚɩɪɚɜɥɟɧɢɟ (1.19, 1.23) ɪɚɜɧɵ:
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a2 a2 |
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22 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
Ɋɚɞɢɭɫ ɤɪɢɜɢɡɧɵ (1.21) ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ: |
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X2 |
X02 g 2t 2 |
3 / 2 |
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(1.48) |
an |
X0 g |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɜɫɟ ɮɨɪɦɭɥɵ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ ɫɩɪɚɜɟɞɥɢɜɵ ɫ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t0 = 0 ɞɨ ɦɨɦɟɧɬɚ ɩɚɞɟɧɢɹ ɬɟɥɚ ɧɚ Ɂɟɦɥɸ t0 d td tɩɚɞ. ɗɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɥɟɝɤɨ ɧɚɣɬɢ ɢɡ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ (1.43), ɩɪɢɧɹɜ ɤɨɨɪɞɢɧɚɬɭ y ɪɚɜɧɨɣ ɧɭɥɸ:
tɩɚɞ |
2H |
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Ɂɚɞɚɱɚ 1.3
(Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ)
Ʌɨɞɤɚ ɩɟɪɟɫɟɤɚɟɬ ɪɟɤɭ ɫ ɩɨɫɬɨɹɧɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɨɞɵ ɫɤɨɪɨɫɬɶɸ ȣɥ , ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɧɚɩɪɚɜɥɟɧɢɸ ɬɟɱɟɧɢɹ ɪɟɤɢ. Ɇɨɞɭɥɶ
ɫɤɨɪɨɫɬɢ ɬɟɱɟɧɢɹ ɪɟɤɢ, ɲɢɪɢɧɚ ɤɨɬɨɪɨɣ d, ɧɚɪɚɫɬɚɟɬ ɨɬ ɛɟɪɟɝɨɜ ɤ ɫɟɪɟɞɢɧɟ ɪɟɤɢ ɩɨ ɩɚɪɚɛɨɥɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ, ɢɡɦɟɧɹɹɫɶ ɨɬ 0 ɞɨ um. ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɥɨɞɤɢ, ɜɪɟɦɹ ɟɟ ɞɜɢɠɟɧɢɹ W , ɚ ɬɚɤɠɟ ɫɧɨɫ ɥɨɞɤɢ l ɜɧɢɡ ɩɨ ɬɟɱɟɧɢɸ ɨɬ ɦɟɫɬɚ ɟɟ ɨɬɩɥɵɬɢɹ ɞɨ ɦɟɫɬɚ ɩɪɢɱɚɥɢɜɚɧɢɹ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɛɟɪɟɝɭ ɪɟɤɢ.
Ɋɟɲɟɧɢɟ
I.ȼɵɛɟɪɟɦ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɭɸ
ɫɛɟɪɟɝɨɦ ɪɟɤɢ, ɢ ɫ ɧɚɱɚɥɨɦ ɜ ɦɟɫɬɟ ɨɬɩɥɵɬɢɹ ɥɨɞɤɢ. Ɉɫɢ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɢ ɫɤɨɪɨɫɬɶ ɬɟɱɟɧɢɹ ɪɟɤɢ u( y) ɢɡɨɛɪɚɠɟɧɵ ɧɚ ɪɢɫ. 1.7.
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Ɋɢɫ. 1.7
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɥɨɞɤɭ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ, ɚ ɛɟɪɟɝɚ ɪɟɤɢ ɩɚɪɚɥɥɟɥɶɧɵɦɢ.
Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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II. Ɂɚɩɢɲɟɦ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɥɨɞɤɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ |
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ɭɫɥɨɜɢɹɦɢ ɡɚɞɚɱɢ: |
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x(0) |
0, y(0) |
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(1.50) |
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ɝɞɟ Xx , Xy – ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɥɨɞɤɢ ɧɚ ɨɫɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ
ɤɨɨɪɞɢɧɚɬ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ (1.26) ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ȣ(t) u( y(t)) ȣɥ (t) ɢɥɢ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:
Xx |
u( y), |
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ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɬɟɱɟɧɢɹ ɪɟɤɢ, ɲɢɪɢɧɚ ɤɨɬɨɪɨɣ d, ɧɚɪɚɫɬɚɟɬ ɨɬ ɛɟɪɟɝɨɜ ɤ ɫɟɪɟɞɢɧɟ ɪɟɤɢ ɩɨ ɩɚɪɚɛɨɥɢɱɟɫɤɨ-
ɦɭ ɡɚɤɨɧɭ, ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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ɝɞɟ a ɢ b – ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɥɢɱɢɧɵ b ɢɫɩɨɥɶɡɭɟɦ ɭɫɥɨɜɢɟ ɡɚɞɚɱɢ:
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ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ Xx (0) a |
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0 ɢ ɫɨɨɬɧɨ- |
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ɲɟɧɢɟ (1.53), ɩɨɥɭɱɢɦ ɜɟɥɢɱɢɧɭ a: |
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III. ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (1.51) ɫ ɭɱɟɬɨɦ (1.52) – (1.54) ɩɪɟɨɛ- |
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ɪɚɡɭɟɬɫɹ ɤ ɜɢɞɭ: |
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ɂɧɬɟɝɪɢɪɭɹ ɭɪɚɜɧɟɧɢɹ (1.55) ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ |
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ɞɥɹ ɤɨɨɪɞɢɧɚɬ ɥɨɞɤɢ (1.50), ɧɚɯɨɞɢɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ: |
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x(t) |
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2um |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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y(t) Xɥt . |
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ɍɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɩɨɥɭɱɚɟɦ, ɢɫɤɥɸɱɚɹ ɜɪɟɦɹ t ɢɡ ɡɚɤɨɧɚ |
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ɞɜɢɠɟɧɢɹ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ (1.56) ɢ (1.57): |
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3Xɥd 2 |
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ɉɨɫɤɨɥɶɤɭ ɜ ɦɨɦɟɧɬ ɩɪɢɱɚɥɢɜɚɧɢɹ y(W ) |
d , ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ |
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W ɥɨɞɤɢ ɪɚɜɧɨ: |
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W |
d |
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(1.59) |
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Xɥ |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɢɫɤɨɦɨɝɨ ɫɧɨɫɚ ɥɨɞɤɢ l ɩɨɥɭɱɢɦ (ɫɦ. 1.58):
l x(W ) |
2um |
d . |
(1.60) |
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Ɂɚɞɚɱɚ 1.4
(ɉɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ)
Ɉɩɪɟɞɟɥɢɬɶ ɮɨɪɦɭ ɬɪɚɟɤɬɨɪɢɢ ɤɚɩɟɥɶ ɞɨɠɞɹ ɧɚ ɛɨɤɨɜɨɦ ɫɬɟɤɥɟ ɬɪɚɦɜɚɹ, ɞɜɢɠɭɳɟɝɨɫɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ ȣ1 , ɜɨ
ɜɪɟɦɹ ɟɝɨ ɬɨɪɦɨɠɟɧɢɹ ɫ ɭɫɤɨɪɟɧɢɟɦ a . Ʉɚɩɥɢ ɞɨɠɞɹ ɩɚɞɚɸɬ ɧɚ ɡɟɦɥɸ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ, ɢ ɫɤɨɪɨɫɬɶ ɢɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɟɦɥɢ ɩɨɫɬɨɹɧɧɚ ɢ ɪɚɜɧɚ ȣ2 .
Ɋɟɲɟɧɢɟ
I. ɇɚɪɢɫɭɟɦ ɱɟɪɬɟɠ ɢ ɢɡɨɛɪɚɡɢɦ ɧɚ ɧɟɦ ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɤɚɩɥɢ ɞɨɠɞɹ ɢ ɬɪɚɦɜɚɹ ɜ ɦɨɦɟɧɬ ɧɚɱɚɥɚ ɬɨɪɦɨɠɟɧɢɹ ɬɪɚɦɜɚɹ (ɪɢɫ. 1.8).
X
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X' |
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ȣ1 |
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a |
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ȣc |
ȣ2 |
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Y'
Y
Ɋɢɫ. 1.8
Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ XY, ɫɜɹɡɚɧɧɭɸ ɫ Ɂɟɦɥɟɣ, ɬɚɤ, ɱɬɨɛɵ ɨɫɶ X ɛɵɥɚ ɧɚɩɪɚɜɥɟɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨ ɜɞɨɥɶ ɭɫɤɨɪɟɧɢɹ ɬɪɚɦɜɚɹ, ɚ ɨɫɶ Y – ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ. ȼɵɛɟɪɟɦ ɬɚɤɠɟ ɜɬɨɪɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ XcYc, ɫɜɹɡɚɧɧɭɸ ɫɨ ɫɬɟɤɥɨɦ ɬɪɚɦɜɚɹ, ɬɚɤ, ɱɬɨɛɵ ɟɟ ɨɫɢ Xc ɢ Yc ɛɵɥɢ ɫɨɧɚɩɪɚɜɥɟɧɵ ɫ ɨɫɹɦɢ X ɢ Y. ȼɪɟɦɹ ɜ ɨɛɟɢɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɢɬɵɜɚɟɦ ɨɬ ɦɨɦɟɧɬɚ ɧɚɱɚɥɚ ɬɨɪɦɨɠɟɧɢɹ ɬɪɚɦɜɚɹ.
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɤɚɩɥɹ ɞɨɠɞɹ ɹɜɥɹɟɬɫɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ, ɩɨɥɨɠɟɧɢɟ ɤɨɬɨɪɨɣ ɜ ɦɨɦɟɧɬ ɧɚɱɚɥɚ ɬɨɪɦɨɠɟɧɢɹ ɬɪɚɦɜɚɹ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɱɚɥɨɦ ɤɨɨɪɞɢɧɚɬ ɫɢɫɬɟɦɵ XcYc.
II. ɂɫɩɨɥɶɡɭɹ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ (1.26), ɡɚɩɢɲɟɦ ɫɤɨɪɨɫɬɶ ȣc ɢ ɭɫɤɨɪɟɧɢɟ ac ɤɚɩɥɢ ɞɨɠɞɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɬɟɤɥɚ ɬɪɚɦɜɚɹ (ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ XcYc):
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ȣ2 |
ȣ1 , |
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ȣ |
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ac a . |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɨɣ ɨɬɫɱɟɬɚ ɡɚɩɢɲɟɦ ɧɚ- |
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ɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɤɚɩɥɢ ɞɨɠɞɹ: |
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(0) |
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0; |
(1.63) |
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0 , y (0) |
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Xcx (0) |
X1 , Xcy (0) |
X2 . |
(1.64) |
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III.Ɂɚɩɢɫɚɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ (1.61) ɢ (1.62)
ɫɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ (1.63) ɢ (1.64) ɩɨɡɜɨɥɹɸɬ ɧɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɤɚɩɥɢ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ:
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°xc |
X1t |
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¯°yc |
X2t. |
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ɍɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɧɚɯɨɞɢɬɫɹ ɢɡ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ ɤɚɩɥɢ ɩɭɬɟɦ ɢɫɤɥɸɱɟɧɢɹ ɢɡ (1.65) ɜɪɟɦɟɧɢ t:
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x |
X1 X2 |
a 2X22 . |
(1.66) |
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Ʉɚɤ ɜɢɞɢɦ, ɬɪɚɟɤɬɨɪɢɹ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ XcYc, ɫɜɹɡɚɧɧɨɣ ɫɨ ɫɬɟɤɥɨɦ ɬɪɚɦɜɚɹ, ɹɜɥɹɟɬɫɹ ɩɚɪɚɛɨɥɨɣ (ɫɦ. ɪɢɫ. 1.9) ɫ ɜɟɪɲɢɧɨɣ ɜ ɬɨɱɤɟ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ:
x' |
X2 |
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X X |
(1.67) |
1 |
1 2 . |
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a |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
X' O
xc xc( yc)
Y'
Ɋɢɫ. 1.9
Ɂɚɞɚɱɚ 1.5
(ɍɪɚɜɧɟɧɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ)
Ʉɨɧɰɵ ɬɜɟɪɞɨɝɨ ɫɬɟɪɠɧɹ MN ɦɨɝɭɬ ɫɜɨɛɨɞɧɨ ɫɤɨɥɶɡɢɬɶ ɩɨ ɫɬɨɪɨɧɚɦ ɩɪɹɦɨɝɨ ɭɝɥɚ MON (ɫɦ. ɪɢɫ. 1.10). ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɬɨɱɤɢ P ɫɬɟɪɠɧɹ, ɤɨɬɨɪɚɹ ɞɟɥɢɬ ɟɝɨ ɧɚ ɱɚɫɬɢ ɞɥɢɧɨɣ ɚ ɢ b.
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɢ ɢɡɨɛɪɚɡɢɦ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɨɫɢ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɸɬ ɫɨ ɫɬɨɪɨɧɚɦɢ ɭɝɥɚ MON (ɫɦ. ɪɢɫ. 1.10).
Y |
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O |
x |
N X |
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Ɋɢɫ. 1.10 |
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɫɬɟɪɠɟɧɶ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɝɨ ɩɨɥɨɠɟɧɢɟ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɨɞɧɨɡɧɚɱɧɨ ɡɚɞɚɟɬɫɹ ɭɝɥɨɦ M(t) ɦɟɠɞɭ ɨɫɶɸ OX ɢ ɫɬɟɪɠɧɟɦ MN.
II. Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ P ɫɬɟɪɠɧɹ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ
ɮɨɪɦɟ (ɫɦ. ɪɢɫ. 1.10): |
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a cosM(t), |
(1.68) |
¯®y(t) |
b sin M(t). |
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Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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ɂɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɬɨɱɤɢ P ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ, ɢɫɤɥɸɱɢɜ ɜɪɟɦɹ ɢɡ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ (1.68).
III. ɉɪɟɨɛɪɚɡɭɹ ɭɪɚɜɧɟɧɢɹ (1.68), ɩɨɥɭɱɚɟɦ:
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cos M(t) sin M(t) |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɩɪɢɧɢɦɚɟɬ |
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ɜɢɞ: |
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ɍɪɚɜɧɟɧɢɟ (1.70) ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɷɥɥɢɩɫɚ ɫ ɩɨɥɭɨɫɹɦɢ, ɫɨɜɩɚɞɚɸɳɢɦɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɫ ɨɫɹɦɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɢ ɪɚɜɧɵɦɢ a ɢ b. ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ a = b, ɷɥɥɢɩɫ ɜɵɪɨɠɞɚɟɬɫɹ ɜ ɨɤɪɭɠɧɨɫɬɶ.
Ɂɚɞɚɱɚ 1.6
(ɍɪɚɜɧɟɧɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ)
ɇɚ ɤɥɢɧɟ ɫ ɭɝɥɨɦ ɩɪɢ ɨɫɧɨɜɚɧɢɢ D, ɪɚɫɩɨɥɨɠɟɧɧɨɦ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɧɚɯɨɞɢɬɫɹ ɫɢɫɬɟɦɚ ɞɜɭɯ ɬɟɥ 1 ɢ 2 (ɫɦ. ɪɢɫ. 1.11), ɫɜɹɡɚɧɧɵɯ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɶɸ, ɩɟɪɟɜɟɲɟɧɧɨɣ ɱɟɪɟɡ ɦɚɥɟɧɶɤɢɣ ɛɥɨɤ, ɨɫɶ ɤɨɬɨɪɨɝɨ ɡɚɤɪɟɩɥɟɧɚ ɜ ɜɟɪɯɧɟɣ ɬɨɱɤɟ ɤɥɢɧɚ. Ɂɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɞɥɹ ɭɫɤɨɪɟɧɢɣ ɤɥɢɧɚ ɢ ɞɜɭɯ ɬɟɥ, ɟɫɥɢ ɬɟɥɨ 2 ɧɟ ɨɬɪɵɜɚɟɬɫɹ ɨɬ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ.
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ. Ɉɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɧɚɩɪɚɜɢɦ ɝɨɪɢɡɨɧɬɚɥɶɧɨ, ɚ ɨɫɶ Y ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ (ɫɦ. ɪɢɫ. 1.11).
Y
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O |
x1 |
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Ɋɢɫ. 1.11 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɬɟɥɚ 1 ɢ 2 ɦɚɬɟɪɢɚɥɶɧɵɦɢ ɬɨɱɤɚɦɢ, ɫɜɹɡɚɧɧɵɦɢ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɶɸ, ɚ ɤɥɢɧ – ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɞɜɢɝɚɬɶɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɜɞɨɥɶ ɨɫɢ X. Ɉɛɨɡɧɚɱɢɦ ɤɨɨɪɞɢɧɚɬɵ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɬɟɥ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ XY – (x1, y1) ɢ (x2, y2), ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ʌɢɧɟɣɧɵɟ ɪɚɡɦɟɪɵ ɛɥɨɤɚ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɞɥɢɧɨɣ ɧɢɬɢ, ɩɨɷɬɨɦɭ ɧɟ ɛɭɞɟɦ ɭɱɢɬɵɜɚɬɶ ɢɯ ɩɪɢ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɣ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɞɥɹ ɤɨɨɪɞɢɧɚɬ ɬɟɥ ɫɢɫɬɟɦɵ.
II. ȼɵɪɚɡɢɦ ɞɥɢɧɭ ɧɢɬɢ l ɱɟɪɟɡ ɜɟɪɬɢɤɚɥɶɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɪɚɡɥɢɱɧɵɯ ɬɨɱɟɤ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ ɬɟɥ:
l |
yɛɥ y1 |
yɛɥ y2 , |
(1.71) |
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sin D |
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ɝɞɟ yɛɥ – ɤɨɨɪɞɢɧɚɬɚ ɛɥɨɤɚ, ɧɟ ɢɡɦɟɧɹɸɳɚɹɫɹ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟ-
ɧɢɹ.
ȿɫɥɢ ɞɥɢɧɭ ɧɚɤɥɨɧɧɨɝɨ ɭɱɚɫɬɤɚ ɧɢɬɢ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɝɨɪɢɡɨɧɬɚɥɶɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɬɟɥ ɫɢɫɬɟɦɵ, ɬɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɞɥɢɧɵ ɧɢ-
ɬɢ ɩɪɢɧɢɦɚɟɬ ɜɢɞ: |
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yɛɥ y2 . |
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cosD |
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III. Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ (1.71) ɢ (1.72) ɞɜɚɠɞɵ ɩɨ ɜɪɟɦɟɧɢ ɢ ɭɱɢ- |
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ɬɵɜɚɹ, |
ɱɬɨ l const ɢ yɛɥ |
const , ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɵɟ ɭɪɚɜɧɟɧɢɹ |
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ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɞɥɹ ɭɫɤɨɪɟɧɢɣ ɬɟɥ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ:
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Ɂɚɞɚɱɚ 1.7
(ɍɪɚɜɧɟɧɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ)
ɋɢɫɬɟɦɚ ɬɟɥ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɛɥɨɤɨɜ ɢ ɞɜɭɯ ɩɨɞɜɟɲɟɧɧɵɯ ɤ ɧɢɦ ɬɟɥ (ɫɦ. ɪɢɫ. 1.12). Ɉɞɢɧ ɢɡ ɛɥɨɤɨɜ ɫɨɫɬɚɜɥɟɧ ɢɡ ɞɜɭɯ ɤɨɚɤɫɢɚɥɶɧɵɯ ɰɢɥɢɧɞɪɨɜ ɫ ɧɟɩɨɞɜɢɠɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɬɨɥɤɚ ɨɫɶɸ, ɢɦɟɸɳɢɯ ɪɚɡɥɢɱɧɵɟ ɪɚɞɢɭɫɵ r ɢ R. ɉɟɪɜɨɟ ɬɟɥɨ ɩɨɞɜɟɲɟɧɨ ɧɚ ɧɢɬɢ, ɧɚɦɨɬɚɧɧɨɣ ɧɚ ɰɢɥɢɧɞɪ ɪɚɞɢɭɫɚ r, ɜɬɨɪɨɣ – ɧɚ ɧɢɬɢ, ɩɪɢɤɪɟɩɥɟɧɧɨɣ ɤ ɨɫɢ ɞɪɭɝɨɝɨ ɛɥɨɤɚ. ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɟ ɜɬɨɪɨɝɨ ɬɟɥɚ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɭɫɤɨɪɟɧɢɟ ɩɟɪɜɨɝɨ ɬɟɥɚ ɪɚɜɧɨ a1. ɇɢɬɢ ɫɱɢɬɚɬɶ ɧɟɪɚɫɬɹɠɢɦɵɦɢ.
Ƚɥɚɜɚ 1. Ʉɢɧɟɦɚɬɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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Ɋɟɲɟɧɢɟ |
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I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɠɟ- |
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ɫɬɤɨ ɫɜɹɡɚɧɧɭɸ ɫ ɩɨɬɨɥɤɨɦ. ɇɚɩɪɚɜɥɟ- R |
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ɧɢɟ ɨɫɟɣ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢ- |
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ɧɚɬ, ɫɜɹɡɚɧɧɨɣ ɫ ɬɟɥɨɦ ɨɬɫɱɟɬɚ, ɩɨɤɚɡɚ- |
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ɧɨ ɧɚ ɪɢɫ. 1.12. |
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ɋɱɢɬɚɟɦ ɬɟɥɚ 1 ɢ 2 ɦɚɬɟɪɢɚɥɶɧɵ- |
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ɦɢ ɬɨɱɤɚɦɢ, ɧɢɬɢ – ɧɟɪɚɫɬɹɠɢɦɵɦɢ. |
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ɉɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɢɬɟɣ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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ɛɥɨɤɨɜ ɧɟɬ. |
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II. ɉɭɫɬɶ ɡɚ ɦɚɥɵɣ ɢɧɬɟɪɜɚɥ ɜɪɟ- |
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ɦɟɧɢ 't ɢɡɦɟɧɟɧɢɟ ɤɨɨɪɞɢɧɚɬɵ ɩɟɪɜɨɝɨ |
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ɬɟɥɚ ɪɚɜɧɨ 'x1 (ɞɥɹ ɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɨɧɨ ɨɩɭɫɤɚɟɬɫɹ). ɉɨ-
ɫɤɨɥɶɤɭ ɧɢɬɶ ɧɟɪɚɫɬɹɠɢɦɚ, ɬɨ ɭɝɨɥ ɩɨɜɨɪɨɬɚ 'M ɰɢɥɢɧɞɪɚ ɪɚɞɢɭɫɨɦ r ɫɜɹɡɚɧ ɫ ɜɟɥɢɱɢɧɨɣ 'x1 ɫɥɟɞɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ:
ǻx1 |
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ɉɪɢ ɷɬɨɦ ɜɬɨɪɨɣ ɰɢɥɢɧɞɪ ɪɚɞɢɭɫɨɦ R ɩɨɜɟɪɧɟɬɫɹ ɧɚ ɬɨɬ ɠɟ |
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ɭɝɨɥ 'M, ɚ ɞɥɢɧɚ ɧɢɬɢ, ɧɚ ɤɨɬɨɪɨɣ ɥɟɠɢɬ ɛɥɨɤ ɫ ɩɨɞɜɟɲɟɧɧɵɦ ɤ |
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ɧɟɦɭ ɬɟɥɨɦ 2, ɢɡɦɟɧɢɬɫɹ ɧɚ ɜɟɥɢɱɢɧɭ: |
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ǻl RǻM . |
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ɂɡɦɟɧɟɧɢɟ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɜɬɨɪɨɝɨ ɛɥɨɤɚ, ɚ ɡɧɚɱɢɬ ɢ ɜɬɨ- |
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ɪɨɝɨ ɬɟɥɚ, ɪɚɜɧɨ: |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (1.74) – (1.76), |
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ɧɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ ɢɡɦɟɧɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɞɜɭɯ ɬɟɥ: |
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ɉɨɞɟɥɢɜ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɱɚɫɬɢ (1.77) ɧɚ ɦɚɥɵɣ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ, ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɞɥɹ ɫɤɨɪɨɫɬɟɣ
ɬɟɥ: |
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Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɩɨɥɭɱɟɧɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɩɨ ɜɪɟɦɟɧɢ, ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɭɸ ɫɜɹɡɶ ɦɟɠɞɭ ɭɫɤɨɪɟɧɢɹɦɢ ɬɟɥ:
30 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
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Ɂɚɞɚɱɚ 1.8 |
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(Ʉɢɧɟɦɚɬɢɤɚ ɩɪɨɫɬɟɣɲɢɯ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ) |
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ɇɚ ɜɚɥ ɪɚɞɢɭɫɚ R, ɡɚɤɪɟɩɥɟɧɧɵɣ ɧɚ |
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ɨɫɢ, ɧɚɦɨɬɚɧɚ ɜɟɪɟɜɤɚ, ɧɚ ɤɨɧɰɟ ɤɨɬɨɪɨɣ |
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ɜɢɫɢɬ ɝɪɭɡ, |
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ɪɢɫ. 1.13). |
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x = x0 + bt2, ɝɞɟ x0 ɢ b – ɩɨɫɬɨɹɧɧɵɟ ɩɨɥɨ- |
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ɠɢɬɟɥɶɧɵɟ ɜɟɥɢɱɢɧɵ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɵɟ |
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ɫɤɨɪɨɫɬɶ Z ɢ ɭɫɤɨɪɟɧɢɟ E ɩɪɨɢɡɜɨɥɶɧɨɣ |
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ɬɨɱɤɢ ɨɛɨɞɚ ɜɚɥɚ, ɦɨɞɭɥɶ ɭɫɤɨɪɟɧɢɹ a, ɟɝɨ |
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ɧɨɪɦɚɥɶɧɭɸ an ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ aW ɩɪɨ- |
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ɟɤɰɢɢ. Ɂɚɩɢɫɚɬɶ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɷɬɨɣ ɬɨɱ- |
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ɤɢ. |
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Ɋɢɫ. 1.13 |
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Ɋɟɲɟɧɢɟ
I. ɇɚɪɢɫɭɟɦ ɱɟɪɬɟɠ ɢ ɢɡɨɛɪɚɡɢɦ ɧɚ ɧɟɦ ɧɚɩɪɚɜɥɟɧɢɟ ɫɤɨɪɨɫɬɢ ȣ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɧɚɩɪɚɜɢɦ ɨɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ (ɪɢɫ. 1.13). Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɫɤɨɪɟɧɢɹ ɢ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ A ɧɚ ɨɛɨɞɟ ɜɚɥɚ ɜɵɛɟɪɟɦ ɩɨɥɹɪɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɫ ɩɨɥɹɪɧɨɣ ɨɫɶɸ Y, ɜ ɤɨɬɨɪɨɣ ɭɝɨɥ M ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬ ɩɨɥɨɠɟɧɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɢ A. ɉɨɫɤɨɥɶɤɭ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɧɟ ɨɝɨɜɚɪɢɜɚɟɬɫɹ ɢɧɨɟ, ɜɟɪɟɜɤɭ ɫɱɢɬɚɟɦ ɧɟɪɚɫɬɹɠɢɦɨɣ ɢ ɱɬɨ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɜɟɪɟɜɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɚɥɚ ɧɟɬ.
II. Ɂɚɩɢɲɟɦ ɡɚɞɚɧɧɵɣ ɜ ɡɚɞɚɱɟ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ:
x x bt 2 . |
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ɉɨɫɤɨɥɶɤɭ ɜɟɪɟɜɤɚ ɧɟɪɚɫɬɹɠɢɦɚ, ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɢɦɟɟɬ ɜɢɞ:
Xx |
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